Then, we recover the result found in [1]}. Now, observe that the sequence \(n\mapsto 1+b^{n}-\frac{n(1-b^{2})}{2}\)  is decreasing. Then there exists \(\widetilde{N}(b)\in \mathbb {N}^*\)  and \(c_0>0\)  such that
\(\inf _{n\in \mathbb {N}^*\atop n\geqslant \widetilde{N}(b)}\Delta _{n}(b)\geqslant c_0>0.\) 
